Methods and Systems for Generating Continuous Surfaces from Polygonal Data

ABSTRACT

Methods and systems for generating surface data from polygonal data are disclosed. The methods and systems receive polygonal data which describe discrete points on an object. The methods and systems analyze and use the data to calculate and define a continuous BREP object which accurately represents the original polygonal object. In some embodiments, the generated BREP is G2 continuous at substantially all points.

CROSS REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of application Ser. No. 12/928,795,filed Dec. 17, 2010, entitled “Methods and Systems for GeneratingContinuous Surfaces from Polygonal Data,” which is incorporated hereinby reference. This application is related to application entitled“Methods and Systems for Generating Continuous Surfaces from PolygonalData,” filed herewith.

BACKGROUND

1. Field

The described technology relates to systems and methods of generatingcontinuous surfaces from data of a polygonal model provided, from, forexample, a 3-D polygonal or Subdivision surface (Sub-D) modeling tool.

2. Description of the Related Technology

Polygonal model data is created with, for example, a 3-D CAD softwaretool by, for example, a designer. The polygonal model or polygonal meshincludes discrete data points describing one or more surfaces orobjects. Polygonal models are convenient for design work at leastbecause they define the surface or object at a degree of detailconvenient for the designer to work with. The shape of the surface orobject is defined by the data points, and the surface between the datapoints is perceived, but is not represented in the data. This allows forthe CAD system to function quicker because of a significantly reduceddata set representing the surface or object, while providing thedesigner enough detail to manipulate to achieve a desired design.

Once the designer has finished the design, the polygonal data can beused, for example, as the basis for manufacturing a physical object orfor generating an image of the designed object. To manufacture theobject, or analyze it, or to generate the image, the mesh data is oftennot sufficient. For a physical object or a realistic image, a BREPobject containing one or more surfaces must be defined. The surfaces arepreferably Non-uniform rational B-spline (NURBS). NURBS is amathematical model commonly used to represent curves and surfaces, whichare either analytic or freeform. A BREP (Boundary REPresentation) may beunderstood to be a geometric and topological representation used, forexample, in CAD applications that links together curves and surfaces toform either a solid (closed volume) or a shell (open).

For accurate CAD models, the continuity of the interior surface and thecontinuity between adjacent surfaces are important characteristics ofthe final product. Mathematically, for adjoining surfaces, continuitygrades of G1 and G2 are defined. Each point on the final BREP object isa point on one or more surfaces. The continuity of the two surfaces at apoint where they touch is characterized as G1 if the surfaces share acommon tangent plane at the point. The continuity of two surfaces at thepoint is characterized as G2 if the surfaces share both a common tangentplane and common curvature at the point. For high quality surfaces G2continuity at most points on the BREP object is highly desirable interms of design aesthetics and manufacturability.

SUMMARY OF CERTAIN INVENTIVE ASPECTS

Various aspects of certain embodiments of methods and systems forgenerating surface data from polygonal data are discussed. The methodsand systems receive polygonal data which describe discrete points on anobject. The methods and systems analyze and use the polygonal data tocalculate and define a continuous BREP object which accuratelyrepresents the object.

One aspect is a method of producing BREP data from electronic polygonaldata. The method includes accessing the polygonal data with a computer,and identifying a plurality of data points, where each identified datapoint corresponds to a non-valence 4 vertex in the polygonal data. Themethod also includes generating one or more continuous surfaces, whereeach generated surface is generated based on points corresponding tovertices in the polygonal data which are near one of the non-valence 4vertices corresponding to an identified data point. The method alsoincludes generating the BREP data based at least in part on thecontinuous surfaces, improving the continuity of the BREP data using thecontinuous surface, and storing the BREP data in a computer readabledata storage.

Another aspect includes a method of producing BREP data from electronicpolygonal data. The method includes accessing the polygonal data with acomputer, where the polygonal data defines a mesh of polygonal datapoints. The method also includes generating a plurality of continuouscurves based on the polygonal data, identifying a plurality of firstdata points corresponding to non-valence 4 vertices in the polygonaldata, and identifying a plurality of second data points corresponding topolygonal data having a third derivative greater than a threshold. Themethod also includes based at least in part on the continuous curves,the first data points, and the second data points, generating aplurality of BREP surface boundaries defining one or more continuousBREP surfaces, generating the BREP data based at least in part on thecontinuous BREP surfaces, and generating one or more additionalcontinuous surfaces, where each additional surface is generated based onpoints corresponding to vertices in the polygonal data which are nearone of the non-valence 4 vertices corresponding to an identifiednon-valence 4 data point. The method also includes improving thecontinuity of the BREP data using the continuous additional surfaces,and storing the improved BREP data in a computer readable data storage.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a graphical illustration of polygonal data.

FIG. 2 is a graphical illustration of continuous BREP object datagenerated from the polygonal data of FIG. 1.

FIG. 3 is a flow chart of a method used to generate the continuous BREPobject data of FIG. 2.

FIG. 4 is a graphical illustration of the polygonal data of FIG. 1 withsurfaces for non-valence 4 vertices.

FIG. 5 is a graphical illustration of continuous curves generated basedon the polygonal data of FIG. 1.

FIGS. 6-13 are graphical illustrations of an embodiment of surfaceboundary generation.

FIGS. 14-16 are graphical illustrations of surface generation.

FIGS. 17-22 are graphical illustrations of surface boundary continuityimprovement.

FIGS. 23-25 are graphical illustrations of continuous BREP object datagenerated by the processes describe with reference to FIGS. 3-22.

DETAILED DESCRIPTION OF CERTAIN INVENTIVE EMBODIMENTS

Various aspects and features of methods and systems are described hereinwith reference to the accompanying drawings, which show certainexemplary embodiments. The described embodiments may be modified invarious ways, without departing from the spirit or scope of the presentinvention. In addition, the described embodiments have multiple featuresand aspects, no single one of which is solely responsible for thedesirable characteristics thereof. Furthermore, no single feature oraspect is essential to practicing the methods and systems describedherein. Furthermore, various features and aspects of the embodiments maybe combined in embodiments not specifically described.

Various inventive aspects of certain embodiments of methods and systemsfor generating BREP object data from polygonal data are discussed. Themethods and systems receive polygonal data which describe discretepoints on a object. The methods and systems analyze and use the data tocalculate and define a BREP which accurately represents the polygonalobject. For example, the methods and systems may receive polygonal data1, such as that represented in FIG. 1 and generate a BREP 2, such asthat represented in FIG. 2. Among other beneficial aspects, the BREPobject defined by the methods and systems may be G2 continuous at all orsubstantially all points on the BREP object, and may be G1 continuous orsubstantially G1 continuous at any non-G2 continuous points. Inaddition, in some embodiments, the BREP object defined by the methodsand systems interpolates all or substantially all of the points in thepolygonal data. In some embodiments, the BREP object defined by themethods and systems includes all or substantially all of the points inthe polygonal data. The methods and systems reliably define BREP objectsfor polygonal data of a wide variety of polygonal data. Whilequadrilateral polygonal data is most efficiently processed, sometriangular polygonal data may be effectively handled. The BREP objectsproduced may contain mostly naturally trimmed (rectangular bounded)surfaces with a layout similar to what might be created by a designer ina CAD system. The flow lines of the resulting BREP curves and surfacesmay closely match the flow lines of the straight edges on the originaldata mesh. Surfaces may be roughly rectangular, two manifold, and/orcurved. Additionally, surfaces can be of any type, for example, NURBSsurface, Bezier surface, Coons surface, Gregory Patch, etc.

A polygonal model or polygonal mesh can be characterized as being, forexample, triangular or quadrilateral. In a predominantly triangularmesh, the polygons defined by the data are generally triangular. In apredominantly triangular mesh most polygons are defined by three datapoints, and each data point is generally a vertex of six triangles. Thenumber of polygons defined by a point is referred to as the valence ofthat point. The valence of that point is also the number of polygonsides or lines which connect to that point. Accordingly, in apredominantly triangular mesh, most of the data points have a valence ofsix. In a predominantly quadrilateral mesh, the polygons defined by thedata are generally quadrilateral. In a predominantly quadrilateral meshmost polygons are defined by four data points, and each data point isgenerally a vertex of four quadrilateral polygons. Therefore, in apredominantly quadrilateral mesh, most of the data points have a valenceof four. FIG. 1 shows polygonal data 1, which is a predominantlyquadrilateral mesh. Certain valence 3 points 11, valence 4 points 12,and valence 5 points 13, are indicated.

In some embodiments, methods and systems receive predominantlyquadrilateral polygonal data representing an object. As indicated above,FIG. 1 shows an example of such data, and FIG. 2 shows an example of acontinuous BREP object 2 corresponding to the surface represented by thepolygonal data 1 shown in FIG. 1. In order to generate the continuousBREP object 2, in some embodiments the polygonal data 1 is analyzed andcontinuous surfaces are formed. The collection of continuous surfacescorresponds to the BREP object 2 represented by the polygonal data 1,where each surface corresponds to a unique portion of the BREP object 2represented by the polygonal data 1. Each of the continuous surfacesforms a portion of the continuous BREP object 2 to be generated. Toimprove the continuity of the generated BREP object 2, the boundariesand vertices of the continuous surfaces are modified. The result is acontinuous BREP object 2 corresponding to the BREP object represented bythe polygonal data 1.

FIG. 3 is a flow chart describing a method of generating BREP objectdata, such as that represented in FIG. 2, from polygonal data, such asthat represented in FIG. 1. The method of FIG. 3 is implemented with acomputer system, which accesses instructions for performing the methodstored on a computer readable medium, such as a memory or data storagedevice. The instructions, when executed by the computer, cause themethod of FIG. 3 to be performed. The polygonal data may be generatedwith a computer CAD system and may be stored in a non-transitorycomputer readable medium, such as a memory or data storage device. Thecomputer system configured to perform or to be used to perform themethod of FIG. 3 accesses the polygonal data to perform the method.

At 10 of FIG. 3, polygonal data 1 is analyzed to identify vertices whichare not valence 4. Several valence 3 vertices 11 and several valence 5vertices 12 are indicated in FIG. 1. Some polygonal data has verticeswith valence higher than 5. In some instances, as a result of the methodused to generate the polygonal data 1, vertices having valence 3 orhaving valence higher than 4 often correspond to points in the polygonaldata 1 with poor continuity. As discussed in more detail below,adjustments may be made to the polygonal data 1 at or near thenon-valence 4 points 14 to improve continuity of the generated BREPobject. In some polygonal data 1, vertices at a boundary of the surfacerepresented may be non-valence 4. In some embodiments, such non-valence4 boundary vertices are excluded from the vertices identified asnon-valence 4 at 10 of FIG. 3.

At 15 of FIG. 3, the non-valence 4 vertices 14, or data pointscorresponding to the non-valence 4 vertices 14 are modified based onnearby vertices to improve continuity across points on the generatedBREP object corresponding to the non-valence 4 vertices 14. In someembodiments, as shown in FIG. 4, continuous surfaces 18 are generatedusing vertices which are near the non-valence 4 vertices 14 whosecontinuity is to be improved. For example, all and/or only the verticesadjacent to a non-valence 4 vertex 14 can be used to generate a G2continuous surfaces 18. In some embodiments, the non-valence 4 vertex 14itself is not used in the BREP object or surface generation. In someembodiments, they are. In some embodiments, vertices used to generatethe continuous surfaces 18 include vertices within 2, 3, 4, 5, or about10 vertices of the non-valence 4 vertex 14. In some embodiments a groupof vertices which substantially surrounds the non-valence 4 vertex 14are included. In some embodiments, vertices used to generate thecontinuous surfaces 18 are limited to vertices within 2, 3, 4, 5, orabout 10 vertices of the non-valence 4 vertex 14. In some embodiments,vertices used to generate the continuous surfaces 18 include about0.001%, about 0.01%, about 0.1%, or about 1% of vertices in thepolygonal data nearest the non-valence 4 vertex 14. In some embodiments,vertices used to generate the continuous surfaces 18 are limited toabout 0.001%, about 0.01%, about 0.1%, or about 1% of vertices in thepolygonal data nearest the non-valence 4 vertex 14.

Some techniques for generating continuous surfaces include variationalsurfacing and/or least squares fitting. Other surface generationtechniques may additionally or alternatively be used, such as otherordered data fitting techniques. In some embodiments, the generatedsurface is a NURBS surface. Alternatively, the generated surface may,for example, be another type of surface, such as a Bezier surface, aCoons surface, a Gregory Patch, etc.) The generated surface may be G2internally continuous. In some embodiments, the surface is not G2internally continuous. Internal surface continuity is generally thecontinuity of a surface inside of the surface boundary.

Once the surface is generated, the non-valence 4 vertex 14 is projectedonto the surface 18 at a projection point where the surface is normal tothe position of the non-valence 4 vertex 14. The projection point isthen used to modify the non-valence 4 vertex 14 in the polygonal data.For example, the non-valence 4 vertex 14 in the polygonal data may bereplaced with data representing the position of the projection point,data representing the position of the projection point may be stored indata representing the BREP surface to be generated, or data representingthe position of the projection point may be stored for later use incalculating the data representing the BREP surface to be generated. Insome embodiments, the normal vector including the projection point andthe unmodified non-valence 4 vertex 14 is stored and used in thegeneration of 3D curves.

At 20 of FIG. 3, the polygonal data is analyzed to identify polygonswhich are not quadrilateral. Some polygonal data has polygons which aretriangles, or have more than 4 sides. In some instances, polygons whichare not quadrilateral may correspond to points in the generated surfacewith poor continuity. As discussed in more detail below, adjustments maybe made to the polygonal data at or near the non-quadrilateral polygonsto improve continuity of the generated BREP object.

At 25 of FIG. 3, the non-quadrilateral polygons are modified based onnearby vertices to improve boundary continuity across points on thegenerated BREP object corresponding to the non-quadrilateral polygons.Vertices included for use to improve the boundary continuity may beincluded for reasons similar to those discussed above regardinginclusion of vertices for improving continuity of the generated BREPdata near non-valence 4 vertices. In some embodiments, continuoussurfaces are generated using vertices which are near thenon-quadrilateral polygons whose continuity is to be improved. Forexample, all the vertices adjacent to a non-quadrilateral polygon can beused to generate a G2 continuous surface. In some embodiments, thevertices of the non-quadrilateral polygon itself are not used in thesurface generation. In some embodiments, they are. In some embodiments,vertices within two vertices of the non-quadrilateral polygon are usedto generate the continuous surface. Such surfaces may be similar tothose shown in FIG. 4 near non-valence 4 vertices.

Once the surface is generated, the vertices of the non-quadrilateralpolygon are projected onto the surface at points where the surface isnormal to the position of each of the vertices. The surface points ofthe projection are then used to modify the vertices of thenon-quadrilateral polygon in the polygonal data. For example, thevertices of the non-quadrilateral polygon in the polygonal data may bereplaced with data representing the positions of the surface points,data representing the positions of the surface points may be stored indata representing the surface to be generated, or data representing thepositions of the surface points may be stored for later use incalculating the data representing the surface to be generated. In someembodiments, the normal vectors including the surface points and theunmodified non-quadrilateral vertices are stored.

At 30 of FIG. 3, the valence 4 vertices 13 are used to create continuouscurves. As shown in FIG. 5, continuous curves 22 including the points ofthe polygonal data vertices are generated. Alternatively, continuouscurves may be generated based on the polygonal data, but not necessarilyincluding the polygonal data points. A technique, such as least squaresfitting, may be used to generate the curves 22 through the valence 4vertices. Other techniques may additionally or alternatively be used. Insome embodiments, the generated curves 22 are NURBS curves. Thegenerated curves 22 may be G2 continuous. In some embodiments, thecurves 22 are not G2 continuous. The curves 22 have end points which areat the boundaries of the polygonal data or which are at non-valence 4vertices 14. On non-valence 4 vertices, modified points from thesurfaces generated in 15 of FIG. 3 and optionally the correspondingsurface normals may be used to generate the curves 22. A plane may bedefined by the center point (of the generated surface) and thecorresponding surface normal at that point. The curves 22 may begenerated so that they are tangent at the common center point so thattheir curve tangents lie substantially on the same plane. The centerpoint is used as an end point of the curves 22. The generated curves 22are used to define the continuous surfaces discussed above.

At 40 of FIG. 3 surface boundaries are defined using the vertices 12having valence 5 and higher. In some embodiments, data pointscorresponding to vertices 12 having valence 5 and higher are used todefine the surface boundaries. As shown in FIG. 6, boundary segments 24are formed. The boundary segments start with the vertices 12 havingvalence 5 and higher and end with the vertices 25 adjacent to thevertices 12 having valence 5 and higher, and include the continuouscurve between the start and end vertices. To further define the surfaceboundaries, the boundary segments 24 are conditionally extended until noboundary segments 24 may be further extended. The rules forconditionally extending the boundary segments may or may not include:

-   -   Boundary segments 24 are extended from valence 4 vertices so as        to continue along the same continuous curve generated in 30 of        FIG. 3.    -   Boundary segments 24 are not extended if the last included        vertex is non-valence 4.    -   Curves along the boundary of the polygonal data form boundary        segments.    -   Boundary segments 24 are not extended if the last included        vertex is in another boundary segment.    -   Boundary segments 24 are not extended if areas of high curvature        change are found

To determine areas of high curvature change, a mathematical test, suchas a threshold on a third derivative at the next or last included vertexmay be used. For example if the third derivative at the last includedvertex is greater than a threshold, the boundary segment 24 may not beextended. Alternatively, if the second derivative at the included vertexis greater than the second derivative at the previous vertex of theboundary segment 24 by a factor greater than a threshold, such as about1.5, about 2, or about 5, the boundary segment 24 may not be extended.

In some embodiments, if a boundary segment 24 is terminated at a vertexof high curvature change, which has a valence of 4, additional boundarysegments (not shown) are generated. The additional boundary segmentsstart from the vertex of high curvature change, and extend in bothdirections along the continuous curve which intersects the continuouscurve along which the terminated segment 24 was being extended. In someembodiments, a boundary segment is additionally started from the vertexof high curvature change and is extended in the direction along thecontinuous curve along which the terminated segment 24 was beingextended. Extending the boundary segments from the data pointscorresponding to vertices of high curvature change may be performedaccording to the rules for extending boundary segments discussed above.

FIGS. 7 and 8 show progressive extensions of the surface boundarysegments 24 started in FIG. 6. FIG. 9 shows the result of the boundarysegments 24 starting with vertices having valence 5 or higher beingfully extended.

At 50 of FIG. 3 surface boundaries are defined using the vertices 11having valence 3. As shown in FIG. 10, boundary segments 26 are formedstarting with the vertices 11 having valence 3 and ending with thevertices 28 adjacent to the vertices 11 having valence 3, and includethe continuous curve between the start and end vertices. To furtherdefine the boundaries, the boundary segments 26 are conditionallyextended in a manner similar to the boundary segments 24 generated fromthe vertices having valence greater than 4. The rules for conditionallyextending the boundary segments 26 may or may not include:

-   -   Boundary segments 26 are extended from valence 4 vertices so as        to continue along the same continuous curve defined in 30 of        FIG. 3.    -   Boundary segments 26 are not extended if the last included        vertex is non-valence 4.    -   Curves along the boundary of the polygonal data form boundary        segments.    -   Boundary segments 26 are not extended if the last included        vertex is in another boundary segment.    -   Boundary segments 26 are not extended if areas of high curvature        change are found

In some embodiments, if a boundary segment 26 is terminated at a vertexof high curvature change, which has a valence of 4, additional boundarysegments (not shown) are generated. The additional boundary segmentsstart from the vertex of high curvature change, and extend in bothdirections along the continuous curve which intersects the continuouscurve along which the terminated segment 26 was being extended. In someembodiments, a boundary segment is additionally started from the vertexof high curvature change and is extended in the direction along thecontinuous curve along which the terminated segment 26 was beingextended.

In some embodiments, data points corresponding to vertices of highcurvature change (for example, at the boundary of a flat surface and asurface with high curvature) are identified in the polygonal dataindependent of the process of extending boundary segments. Theidentified data points may be subsequently used as starting points forgenerating boundary segments using a process similar to those describedabove.

In some embodiments, the surface boundary segments are iterativelyextended, such that during each iteration, each of the boundary segmentsbeing extended is extended to include an additional point which at leastone of corresponds to a vertex of the polygonal data and is included onone of the continuous curves generated at 30 of FIG. 3. In someembodiments, all of or only the surface boundaries extended fromvertices 12 having valence 5 and higher are extended during aniteration. In some embodiments, all of or only the surface boundariesextended from vertices 11 having valence 3 are extended during aniteration. In some embodiments, all of or only the surface boundariesextended from vertices of high curvature change are extended during aniteration.

In some embodiments, all of or only the surface boundaries extended fromvertices 12 having valence 5 and higher are extended before surfaceboundaries extended from other vertices. In some embodiments, all of oronly the surface boundaries extended from vertices 11 having valence 3are extended before surface boundaries extended from other vertices. Insome embodiments, all of or only the surface boundaries extended fromvertices of high curvature change are extended before surface boundariesextended from other vertices.

Alternatively, all surface boundaries or surface boundaries started fromvertices 12 having valence 5 and higher, vertices 11 having valence 3,and vertices of high curvature change may be extended during a singleiteration.

In some embodiments, data points of the non-quadrilateral polygons areused as starting points from which boundary segments are extended. Aprocess for extending the boundary segments form data points of thenon-quadrilateral polygons may be similar to those discussed above.

FIG. 11 shows extensions of the surface boundary segments 26 formed inFIG. 10. FIG. 12 shows the result of the boundary segments 26 startingwith vertices 11 having valence 3 being fully extended. FIG. 13 showsthe result of the boundary segments 24 starting with vertices 12 havingvalence 5 or higher being fully extended and the boundary segments 26starting with vertices 11 having valence 3 being fully extended. Alsoshown in FIG. 13, with the addition of the surface boundary segments 28defined by the continuous curves generated in 30 which include boundaryvertices of the polygonal data, the surface boundaries 32 are fullyformed and define the boundaries for the surfaces to be generated forsurface locations 33 of the surface.

At 60 of FIG. 3, continuous surfaces are generated for each area 33 ofthe mesh surrounded by boundary segments. Portions 34 of the continuouscurves generated at 30 of FIG. 3 which are within the bounded areas 33are used to generate the surface for each area 33. Such a set of curveportions 34 is shown in FIG. 14, which shows curve portions 34 for asurface location 36 between the eyes, as shown in FIG. 13. FIG. 15 showsthe vertices 38 in the surface location 36 along with tangent vectors 42at each vertex. The tangent vectors 42 represent the slopes of thecontinuous curve portions 34 at the vertices 38.

The curve portions 34, as shown in FIG. 14, and the tangent vectors 42,as shown in FIG. 15 may be used by a surface generation technique todefine a continuous surface. FIG. 16 is a representation of the surface44, which is a continuous surface 44 generated from the curve portions34 of FIG. 14 and the tangent vectors 42 of FIG. 15. Some techniques forgenerating the continuous surface 44 include variational surfacingand/or least squares fitting. Other surface generation techniques mayadditionally or alternatively be used, such as another unordered datafitting technique. In some embodiments, the generated surface is a NURBSsurface. The surface may be another type of surface. The generatedsurface may be G2 continuous. In some embodiments, the surface is not G2continuous.

In order to generate the continuous BREP object corresponding to all ofthe polygonal data, a continuous surface is generated for each area 33surrounded by boundary segments. The collection of continuous surfacesforms the continuous BREP corresponding to all of the polygonal data.Because the curves used to generate the surfaces are continuous thesurfaces are internally continuous. In addition, the surface boundariesare continuous where the vertices on the boundary have valence 4. Thisis the case because the curves which cross the boundaries at verticeshaving valence 4 are continuous, as discussed above with reference to 30of FIG. 3. Therefore, to improve the continuity of the entire generatedBREP, at 70 of FIG. 3, the boundaries of the continuous surfaces nearnon-valence 4 vertices are analyzed and modified where needed.

At some boundaries between a non-valence 4 vertex and the adjacentvalence 4 vertex along the boundary, an unacceptable discontinuity mayexist. FIG. 17 shows a boundary 46 between two surfaces 48 which has avalence 5 vertex 52. To determine and address the discontinuity, newknots 54 are added to the surfaces 48 sharing the boundary. The newknots 54 are added, as shown in FIG. 18 so as to be between thepreviously existing vertices 56. In some embodiments, the added knots 54are located at a middle point between the previously existing vertices56. While the discontinuity is concentrated near the non-valence 4vertex 52, new knots 54 are added around the entire perimeter of eachsurface 48 to avoid high aspect ratio portions of the surface data nearthe non-valence 4 surface vertex 52. Using a process similar to thatdiscussed above with reference to FIG. 14, continuous curves 58 aregenerated from the new vertices 56 on each surface 48, as shown in FIG.18. In addition, using a process similar to that discussed above withreference to FIG. 15, tangent vectors 62 are determined at each of theold and new vertices, as shown in FIG. 19.

As shown in FIG. 19, the new vertex 64 on the boundary nearest thevalence 5 vertex has two tangent vectors 66. The tangent vectors 66 arefrom the continuous curves of both surfaces 48 sharing the boundary. Asshown in FIG. 19, the two tangent vectors 66 are not parallel. Thisindicates a discontinuity between the curves and between the surfaces atthe new vertex 64. The angle between the two tangent vectors 66 is ameasure of the discontinuity. To improve the continuity of the boundary,the two tangent vectors 66 may be modified so as to be equal andopposite, for example, by modifying each tangent vector 66 by a commonamount. In some embodiments, the tangent vectors 66 are modified bydifferent amounts. In some embodiments, the location of one or more ofthe new and/or old vertices is changed to reduce the angle between thetangent vectors 66. In some embodiments, the continuity is improved onlyif the angle is greater than a threshold, for example, 1 degree or 1percent. FIG. 20 shows that, after modification, the tangent vectors 66of the new vertex 64 are equal and opposite. Accordingly, a curvethrough the new vertex 64 from one surface to the other is G1continuous. In some embodiments, the new and/or old vertices near thenew vertex 64 are modified so that the curve through the new vertex 64from one surface to the other is G2 continuous. FIG. 21 shows thesurfaces 48 after the continuity between the surfaces 48 near thevalence 5 vertex 52 has been improved. Once the tangent vectors aresatisfactory, each of the affected surfaces is regenerated based on datawhich includes the new vertices and tangent vectors.

In some embodiments, improvement of the continuity near the non-valence4 vertex is repeated. For example, if the angle between the tangentvectors 66 prior to modification is greater than a threshold, theimprovement process may be repeated. FIG. 22 shows the surfaces 48 afterthe continuity between the surfaces near the valence 5 vertex 52 hasbeen improved a second time. As shown, the second improvement processadded new vertices 68 and continuous curves 72 including the newvertices 68.

FIG. 23 shows the continuous surface 74 generated by the method of FIG.3, with the surface boundaries 76. FIG. 24 shows the continuous surface74 generated by the method of FIG. 3, with the continuous curves 78 andvertices 82.

The generated BREP 74 may contain a NURBS surface, and may be G2continuous at all or substantially all points, and may be G1 continuousat points which are not G2 continuous. Once data representing thecontinuous BREP object 74 is generated, the data may be stored in anon-transitory computer readable medium, such as a memory storagedevice. The data may be used to generate an electronic or printed imageof the continuous BREP object. The data may also be used to generate aphysical representation or instructions for generating a physicalrepresentation of the continuous BREP.

FIG. 25 shows a zebra stripe analysis 82 of the continuous BREP 74generated by the method of FIG. 3. As shown the zebra stripe analysisindicates that the continuity of the continuous BREP generated by themethod of FIG. 3 is excellent.

The various aspects, processes, actions may be performed sequentially orin parallel. For example, a system capable of parallel processing maydivide certain procedures among the available processing devices.

While various aspects, processes, actions, and systems have beendescribed as being included in the embodiments discussed, the variousaspects, processes, actions, and systems can be practiced with certainmodifications. For example, the sequential order of the various aspects,processes, and actions may be modified. In addition, certain aspects,processes, and actions may be omitted, and other aspects, processes, andactions may be added.

1. A method of producing BREP data from electronic polygonal data, themethod comprising: accessing the polygonal data with a computer, thepolygonal data defining a mesh of polygonal data points; generating aplurality of continuous curves based on the polygonal data; identifyinga plurality of first data points corresponding to non-valence 4 verticesin the polygonal data; identifying a plurality of second data pointscorresponding to polygonal data having a third derivative greater than athreshold; based at least in part on the continuous curves, the firstdata points, and the second data points, generating a plurality of BREPsurface boundaries defining one or more continuous BREP surfaces;generating the BREP data based at least in part on the continuous BREPsurfaces; generating one or more additional continuous surfaces, eachadditional surface being generated based on points corresponding tovertices in the polygonal data which are near one of the non-valence 4vertices corresponding to an identified non-valence 4 data point;improving the continuity of the BREP data using the continuousadditional surfaces; and storing the improved BREP data in a computerreadable data storage.
 2. The method of claim 1, wherein the continuouscurves include the points of the mesh.
 3. The method of claim 1, whereinthe continuous curves are G2 continuous.
 4. The method of claim 1,wherein the BREP surface boundaries intersect one or more of the curves.5. The method of claim 1, wherein the surface boundaries are G2continuous at points corresponding to valence 4 vertices in thepolygonal data.
 6. The method of claim 1, wherein improving thecontinuity of the BREP data comprises modifying an identified first datapoint based on one of the additional continuous surfaces.
 7. The methodof claim 1, wherein improving the continuity of the BREP data comprisesadding a plurality of knots to the BREP data based on one of theadditional continuous surfaces.
 8. The method of claim 1, whereingenerating the BREP surface boundaries comprises extending surfaceboundary segments from the identified first data points.
 9. The methodof claim 1, wherein generating the BREP surface boundaries comprisesextending surface boundary segments from the second data points.
 10. Themethod of claim 1, wherein each first vertex having valence ncorresponds to a point on the BREP surface boundary of n of thecontinuous BREP surfaces, wherein n is an integer.
 11. A method ofproducing BREP data from electronic polygonal data, the methodcomprising: accessing the polygonal data with a computer; identifying aplurality of data points, each identified data point corresponding to anon-valence 4 vertex in the polygonal data; generating one or morecontinuous surfaces, each generated surface being generated based onpoints corresponding to vertices in the polygonal data which are nearone of the non-valence 4 vertices corresponding to an identified datapoint; generating the BREP data based at least in part on the continuoussurfaces; improving the continuity of the BREP data using the continuoussurface; and storing the BREP data in a computer readable data storage.12. The method of claim 11, wherein each continuous surface is G2continuous.
 13. The method of claim 11, wherein each continuous surfaceis generated based on points corresponding to vertices which areadjacent to the corresponding non-valence 4 vertex.
 14. The method ofclaim 11, wherein each continuous surface is generated based on pointscorresponding to vertices which substantially surround the correspondingnon-valence 4 vertex.
 15. The method of claim 11, wherein improving thecontinuity of the BREP data comprises modifying each identified pointbased on the corresponding continuous surface.
 16. The method of claim12, wherein modifying each identified point comprises projecting theidentified point onto the corresponding continuous surface at aprojection point.
 17. The method of claim 16, wherein each continuoussurface is normal to the corresponding identified point at theprojection point.
 18. The method of claim 11, wherein improving thecontinuity of the BREP data comprises adding a plurality of knots to theBREP data based on each continuous surface.
 19. The method of claim 18,wherein improving the continuity of the BREP data comprises modifyingeach identified point based on the corresponding continuous surface, andwherein the knots are added between each modified point and other pointscorresponding to adjacent points.
 20. The method of claim 18, furthercomprising generating a plurality of surface boundaries defining one ormore additional continuous surfaces, wherein generating the BREP data isbased at least in part on the additional surfaces, and wherein the knotsare added on the additional surfaces.
 21. The method of claim 20,further comprising improving the continuity of the knots added on theboundaries defining the additional surfaces.
 22. The method of claim 21,wherein improving the continuity of the knots added on the boundariescomprises: generating a continuous curve on each of the additionalsurfaces which share a boundary, wherein each continuous curve on eachadditional surface includes knots added on the surface and has an addedknot on the shared boundary as an end point; determining a tangent foreach of the added knots on each curve, wherein the added knot on theshared boundary has a first tangent from a first curve on a firstadditional surface and a second tangent from a second curve on a secondadditional surface; modifying the first and second tangents so as to beparallel; and regenerating the first and second additional surfacesbased at least in part on the added knots and the tangents of the addedknots.
 23. The method of claim 11, further comprising generating aplurality of surface boundaries defining one or more additionalcontinuous surfaces, wherein the identified points correspond to pointson the surface boundaries.